Average Error: 16.5 → 14.2
Time: 11.3s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.7963576877929056 \cdot 10^{-230}:\\ \;\;\;\;\left(x + \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{t}\right) \cdot \left(\sqrt[3]{y} \cdot z\right)\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;y \le 1.649048289790591 \cdot 10^{-41}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \sqrt{y} \cdot \left(\sqrt{y} \cdot \frac{z}{t}\right)}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -8.7963576877929056 \cdot 10^{-230}:\\
\;\;\;\;\left(x + \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{t}\right) \cdot \left(\sqrt[3]{y} \cdot z\right)\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\

\mathbf{elif}\;y \le 1.649048289790591 \cdot 10^{-41}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \sqrt{y} \cdot \left(\sqrt{y} \cdot \frac{z}{t}\right)}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\

\end{array}
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (x + ((double) (((double) (y * z)) / t)))) / ((double) (((double) (a + 1.0)) + ((double) (((double) (y * b)) / t))))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((y <= -8.796357687792906e-230)) {
		VAR = ((double) (((double) (x + ((double) (((double) (((double) cbrt(y)) * ((double) (((double) cbrt(y)) / t)))) * ((double) (((double) cbrt(y)) * z)))))) * ((double) (1.0 / ((double) (a + ((double) (1.0 + ((double) (y * ((double) (b / t))))))))))));
	} else {
		double VAR_1;
		if ((y <= 1.649048289790591e-41)) {
			VAR_1 = ((double) (((double) (x + ((double) (((double) (y * z)) / t)))) / ((double) (((double) (a + 1.0)) + ((double) (((double) (y * b)) / t))))));
		} else {
			VAR_1 = ((double) (((double) (x + ((double) (((double) sqrt(y)) * ((double) (((double) sqrt(y)) * ((double) (z / t)))))))) / ((double) (a + ((double) (1.0 + ((double) (y * ((double) (b / t))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.5
Target13.1
Herbie14.2
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -8.7963576877929056e-230

    1. Initial program 18.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

    2. Simplified16.6

      \[\leadsto \color{blue}{\frac{x + y \cdot \frac{z}{t}}{a + \left(1 + y \cdot \frac{b}{t}\right)}}\]
    3. Using strategy rm

    4. Applied div-inv16.6

      \[\leadsto \color{blue}{\left(x + y \cdot \frac{z}{t}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}}\]
    5. Using strategy rm

    6. Applied associate-*r/18.1

      \[\leadsto \left(x + \color{blue}{\frac{y \cdot z}{t}}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]
    7. Using strategy rm

    8. Applied associate-/l*16.5

      \[\leadsto \left(x + \color{blue}{\frac{y}{\frac{t}{z}}}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]
    9. Using strategy rm

    10. Applied div-inv16.5

      \[\leadsto \left(x + \frac{y}{\color{blue}{t \cdot \frac{1}{z}}}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]

    11. Applied add-cube-cbrt16.7

      \[\leadsto \left(x + \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{t \cdot \frac{1}{z}}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]

    12. Applied times-frac17.5

      \[\leadsto \left(x + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z}}}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]

    13. Simplified17.5

      \[\leadsto \left(x + \color{blue}{\left(\frac{\sqrt[3]{y}}{t} \cdot \sqrt[3]{y}\right)} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z}}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]

    14. Simplified17.4

      \[\leadsto \left(x + \left(\frac{\sqrt[3]{y}}{t} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{\left(z \cdot \sqrt[3]{y}\right)}\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]

    if -8.7963576877929056e-230 < y < 1.649048289790591e-41

    1. Initial program 3.2

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

    if 1.649048289790591e-41 < y

    1. Initial program 27.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

    2. Simplified20.3

      \[\leadsto \color{blue}{\frac{x + y \cdot \frac{z}{t}}{a + \left(1 + y \cdot \frac{b}{t}\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt20.4

      \[\leadsto \frac{x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \frac{z}{t}}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]

    5. Applied associate-*l*20.4

      \[\leadsto \frac{x + \color{blue}{\sqrt{y} \cdot \left(\sqrt{y} \cdot \frac{z}{t}\right)}}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]

    6. Simplified20.4

      \[\leadsto \frac{x + \sqrt{y} \cdot \color{blue}{\left(\frac{z}{t} \cdot \sqrt{y}\right)}}{a + \left(1 + y \cdot \frac{b}{t}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.7963576877929056 \cdot 10^{-230}:\\ \;\;\;\;\left(x + \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{t}\right) \cdot \left(\sqrt[3]{y} \cdot z\right)\right) \cdot \frac{1}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;y \le 1.649048289790591 \cdot 10^{-41}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \sqrt{y} \cdot \left(\sqrt{y} \cdot \frac{z}{t}\right)}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))